Lemma F.1 (Binomial Confidence Interval [1]}) 

Let \(c\)  be a realization of a random variable \(C \sim Binomial(n,p)\) .
Let \(B_{a,b} \sim Beta(a,b)\)  be a random variable
representing the Beta distribution with shape parameters \(a,b\)  and let
\(F_{B(a,b)}\)  be its CDF. Then, for any \(\alpha \in (0,1)\) ,
\( \operatornamewithlimits{\mathbb {P}}\left( F_{B(c,n-c+1)}^{-1} (\alpha ) \leqslant p \right) \geqslant 1 - \alpha . \) 

